Modular invariants of pseudo-reflection groups

Abstract

In this chapter, we discuss the modular case of invariant theory. We proved in Chapter 18 that, in the nonmodular case, a finite group G ⊂ GL(V) is a pseudo-reflection group if and only if the ring of invariants S(V)^G is a polynomial algebra. This result is only partly true in the modular case. In §19–1, it will be shown that, if S(V)^G is polynomial, then G ⊂ GL(V) is a pseudo-reflection group. However, the converse is not true. There are modular pseudo-reflection groups whose ring of invariants are not polynomial. Indeed, the ring of invariants of a pseudo-reflection group can be quite complex. On the other hand, if we pass from the ordinary invariants of a pseudo-reflection group to its “generalized invariants”, then this invariant theory is much better behaved. These generalized invariants will be discussed in § 19–2 and § 19–3.